Moduli Spaces: New Directions

代数几何研究变化：由多项式方程组定义的形状。参数的多项式约束在数学（例如数论中的丢芬图方程，如著名的费马方程）和更广泛的应用中都很常见，从物理学到工程和代数统计，代数变量被用来描述各种构型空间（甚至宇宙的形状！）。多项式固有的刚性，特别是与拓扑和微分几何中允许的更灵活的变换相比，使得应用来自代数（环和模块的研究）和偶数理论的广泛技术成为可能，例如约模素数。代数几何中最基本的问题之一就是学习如何对代数变量的几何性质进行分类和区分。除了离散参数（多数为拓扑性质）外，代数变量还具有连续参数（称为“模”）。例如，黎曼曲面（如球体或环面）通过孔的数量（属）在拓扑上进行分类。黎曼已经知道它们也有3g-3连续复参数（g是属）。值得注意的是，这些参数可以被解释为“模空间”上的坐标：编码给定拓扑形状的所有可能几何结构的主空间。PI是低维变量（曲线和曲面）的模空间的专家。该项目将侧重于开发新的技术和方法来研究它们。第一种方法是基于凸几何，它研究由线性不等式定义的形状（如多边形和多面体）。一个可以追溯到牛顿的思想是捕捉一个特殊函数系统的渐近行为，以构造一个弯曲几何的多面体近似。另一种方法是线性化。派生类将几何线性化，就像切线近似函数图一样。PI旨在理解几个模空间的派生范畴。最后，模空间可以通过模素数化简来研究。为此，模空间必须配备额外的数据来同时参数化复杂和有限的几何形状。除了推进代数几何之外，拟议的研究计划还通过招聘，培训和监督研究生和本科生，并为他们的研究，教学和专业发展提供资源，为多样化，具有全球竞争力的STEM劳动力的发展做出贡献。特别是，该项目包括三个专门针对REU（本科生研究经验）的问题以及几个博士论文问题。PI还将继续开发研究生和本科生课程，包括旨在向更广泛的公众介绍数学和科学的课程。PI将研究代数几何中的几个基本问题，重点是局部系统、曲线和曲面的紧模空间。PI先前已经开发了代数环面中子变种的经典“交换”热带化到模空间的应用。最近，PI和他的研究生Vogiannou将经典热带几何推广到一个“非交换”的环境，PI提出利用这一推广来构造穿孔黎曼球上局部系统模空间的几何上有意义的紧化。在与Castravet的合作中，PI打算验证Orlov和Kuznetsov关于稳定有理曲线（及相关空间）的模空间的衍生范畴的猜想，并应用该描述来寻找其非交换变形。PI还打算利用Halpern-Leistner发展的窗口理论来解决一些关于环面和对数法诺变种派生范畴的开放问题。在与Freixas i Montplet的合作中，PI提议系统地发展曲线模空间的Arakelov几何，并将其应用于数论问题。PI与Urzua合作提出了几个关于曲面模的新项目，包括具有QHD奇点的曲面的模空间的构造（允许具有有理同调圆盘Milnor纤维的光滑）和循环商奇点变形的总空间的反翻转族，方向是回答Kollar问题。此外，还有三个为reu量身定制的项目：环面和椭圆直纹曲面上的Seshadri常数，以及超平面排列补的对数正则紧化方程。

Algebraic geometry studies varieties: shapes defined by systems of polynomial equations. Polynomial constraints on parameters are quite common both in mathematics (for example, Diophantine equations in number theory, such as the famous Fermat equation) and in broader applications ranging from physics, where algebraic varieties are used to describe various configuration spaces (and even the shape of the universe!), to engineering and algebraic statistics. The inherent rigidity of polynomials, especially compared to more flexible transformations allowed in topology and differential geometry, makes possible to apply a wide range of techniques coming from algebra (the study of rings and modules) and even number theory, for example reduction modulo prime numbers. One of the most basic problems in algebraic geometry is to learn how to classify and distinguish geometric properties of algebraic varieties. In addition to discrete parameters, mostly of topological nature, algebraic varieties also have continuous parameters called "moduli". For example, Riemann surfaces (such as a sphere or a torus) are classified topologically by the number of holes (the genus). It was known already to Riemann that they also have 3g-3 continuous complex parameters (g is the genus). Quite remarkably, these parameters can be interpreted as coordinates on the "moduli space": the master space that encodes all possible geometric structures of a given topological shape. The PI is an expert on moduli spaces of low-dimensional varieties (curves and surfaces). The project will focus on developing new techniques and approaches to their study. The first approach is based on convex geometry, which studies shapes defined by linear inequalities (such as polygons and polyhedra). An idea that goes back to Newton is to capture asymptotic behavior of a distinguished system of functions to construct a polyhedral approximation of a curved geometry. Another approach is linearization. Derived categories linearize geometries just like tangent lines approximate graphs of functions. The PI aims to understand derived categories of several moduli spaces. Finally, moduli spaces can be studied using number theory by reducing them modulo prime numbers. To this end, moduli spaces have to be equipped with extra data to parametrize complex and finite geometries simultaneously. In addition to advancing algebraic geometry, the proposed research program contributes to the development of a diverse, globally competitive STEM workforce through recruiting, training, and supervising of graduate and undergraduate students and providing resources for their research, teaching, and professional development. In particular, the project includes three problems specifically tailored for REU (Research Experience for Undergraduates) as well as several PhD thesis problems. The PI will also continue to develop graduate and undergraduate courses, including courses designed to introduce a broader public to mathematics and science.The PI will work on several fundamental problems in algebraic geometry with a focus on compact moduli spaces of local systems, curves and surfaces. The PI has previously developed applications of classical "commutative" tropicalization of subvarieties in algebraic tori to moduli spaces. Recently, classical tropical geometry was extended by the PI and his graduate student Vogiannou to a "non-commutative" setting and the PI proposes to use this generalization to construct a geometrically meaningful compactification of the moduli space of local systems on a punctured Riemann sphere. In collaboration with Castravet, the PI intends to verify a conjecture of Orlov and Kuznetsov on derived category of the moduli space of stable rational curves (and related spaces) and to apply this description to find its non-commutative deformations. The PI also intends to use the theory of windows as developed by Halpern-Leistner to solve some open problems about derived categories of toric and log Fano varieties. In collaboration with Freixas i Montplet, the PI proposes to systematically develop Arakelov geometry of the moduli space of curves and to apply it to questions in number theory. The PI proposes several new projects on moduli of surfaces in collaboration with Urzua, including constructions of moduli spaces of surfaces with QHD singularities (admitting smoothings with rational homology disc Milnor fiber) and families of antiflips of total spaces of deformations of cyclic quotient singularities in the direction of answering a question of Kollar. In addition there are three projects tailored for REUs: Seshadri constants on toric and elliptic ruled surfaces, and equations of log canonical compactifications of complements of hyperplane arrangements.